The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method [26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that are not coupled, which has never been done to our knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS.

Large scale linear time invariant (LTI) systems arise in many physical and technical fields. An approximation, e.g. with model order reduction techniques, of this large systems is crucial for a cost efficient simulation. In this thesis we focus on a model order reduction method based on modal approximation, where the LTI system is projected onto the left and right eigenspaces corresponding to the dominant poles of the system. These dominant poles are related to the most dominant parts of the residue expansion of the transfer function and usually form a small subset of the eigenvalues of the system matrices. The computation of this dominant poles can be a formidable task, since they can lie anywhere inside the spectrum and the corresponding left eigenvectors have to be approximated as well. We investigate the subspace accelerated dominant pole algorithm and the two-sided and alternating Jacobi-Davidson method for this modal truncation approach. These methods can be seen as subspace accelerated versions of certain Rayleigh quotient iterations. Several strategies that admit an efficient computation of several dominant poles of single-input single-output LTI systems are examined. Since dominant poles can lie in the interior of the spectrum, we discuss also harmonic subspace extraction approaches which might improve the convergence of the methods. Extentions of the modal approximation approach and the applied eigenvalue solvers to multi-input multi-output are also examined. The discussed eigenvalue algorithms and the model order reduction approach will be tested for several practically relevant LTI systems.

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Many methods for computing eigenvalues of a large sparse matrix involve shift-invert transformations which require the solution of a shifted linear system at each step. This thesis deals with shift-invert iterative techniques for solving eigenvalue problems where the arising linear systems are solved inexactly using a second iterative technique. This approach leads to an inner-outer type algorithm. We provide convergence results for the outer iterative eigenvalue computation as well as techniques for efficient inner solves. In particular eigenvalue computations using inexact inverse iteration, the Jacobi-Davidson method without subspace expansion and the shift-invert Arnoldi method as a subspace method are investigated in detail. A general convergence result for inexact inverse iteration for the non-Hermitian generalised eigenvalue problem is given, using only minimal assumptions. This convergence result is obtained in two different ways; on the one hand, we use an equivalence result between inexact inverse iteration applied to the generalised eigenproblem and modified Newton's method; on the other hand, a splitting method is used which generalises the idea of orthogonal decomposition. Both approaches also include an analysis for the convergence theory of a version of inexact Jacobi-Davidson method, where equivalences between Newton's method, inverse iteration and the Jacobi-Davidson method are exploited. To improve the efficiency of the inner iterative solves we introduce a new tuning strategy which can be applied to any standard preconditioner. We give a detailed analysis on this new preconditioning idea and show how the number of iterations for the inner iterative method and hence the total number of iterations can be reduced significantly by the application of this tuning strategy. The analysis of the tuned preconditioner is carried out for both Hermitian and non-Hermitian eigenproblems. We show how the preconditioner can be implemented efficiently and illustrate its performance using various numerical examples. An equivalence result between the preconditioned simplified Jacobi-Davidson method and inexact inverse iteration with the tuned preconditioner is given. Finally, we discuss the shift-invert Arnoldi method both in the standard and restarted fashion. First, existing relaxation strategies for the outer iterative solves are extended to implicitly restarted Arnoldi's method. Second, we apply the idea of tuning the preconditioner to the inner iterative solve. As for inexact inverse iteration the tuned preconditioner for inexact Arnoldi's method is shown to provide significant savings in the number of inner solves. The theory in this thesis is supported by many numerical examples.

Large scale linear time invariant (LTI) systems arise in many physical and technical fields. An approximation, e.g. with model order reduction techniques, of this large systems is crucial for a cost efficient simulation. In this thesis we focus on a model order reduction method based on modal approximation, where the LTI system is projected onto the left and right eigenspaces corresponding to the dominant poles of the system. These dominant poles are related to the most dominant parts of the residue expansion of the transfer function and usually form a small subset of the eigenvalues of the system matrices. The computation of this dominant poles can be a formidable task, since they can lie anywhere inside the spectrum and the corresponding left eigenvectors have to be approximated as well. We investigate the subspace accelerated dominant pole algorithm and the two-sided and alternating Jacobi-Davidson method for this modal truncation approach. These methods can be seen as subspace accelerated versions of certain Rayleigh quotient iterations. Several strategies that admit an efficient computation of several dominant poles of single-input single-output LTI systems are examined. Since dominant poles can lie in the interior of the spectrum, we discuss also harmonic subspace extraction approaches which might improve the convergence of the methods. Extentions of the modal approximation approach and the applied eigenvalue solvers to multi-input multi-output are also examined. The discussed eigenvalue algorithms and the model order reduction approach will be tested for several practically relevant LTI systems.

The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup<p>diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method<p>[26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular<p>aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that<p>are not coupled, which has never been done to our<p>knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished